On my copy of LeafGreen, I bred myself a Tyrogue. It has a Quiet nature. I already have both Hitmonlee and Hitmonchan, so I want Hitmontop. What would be the best way for me to level it up so that the attack and defense stats remain the same?

Quiet nature raises Special Attack and lowers Speed, so it does not affect the Attack stat.

Just try using vitamins (specifically Iron and Protein in this case) and you can always save before evolving it, then try and evolve it. If it does not evolve into Hitmontop, you can reset, add more into the stats and try again. Hope that helped you.